Problem 14 - Algebra Form and Function

Question

Are the two functions the same function? $$ h(t)=t^{2}-t(t-1) \text { and } g(t)=t $$

Step-by-Step Solution

Answer

Answer: Yes, the functions h(t) and g(t) are the same function since their simplified expressions are equivalent (h(t) = t = g(t)).

Step 1: Simplify h(t)

First, we simplify the given function h(t). We have: $$ h(t) = t^2 - t(t-1) $$ Apply the distributive property to expand the second term: $$ h(t) = t^2 - (t^2 - t) $$ Now, simplify the expression by combining like terms: $$ h(t) = t^2 - t^2 + t $$ $$ h(t) = t $$ The simplified form of h(t) is just t.

Step 2: Simplify and compare g(t)

The given function g(t) is already simplified: $$ g(t) = t $$ Now, compare the simplified forms of h(t) and g(t): $$ h(t) = t = g(t) $$ Since the simplified expressions of h(t) and g(t) are equivalent, the two functions are indeed the same function.

Key Concepts

Distributive Property

The distributive property is a fundamental concept that often helps simplify mathematical expressions. In the context of functions, it allows us to expand expressions by distributing terms over a sum or difference. For example, consider the function \( h(t) = t^2 - t(t-1) \). Here, the distributive property can be applied to the second term \( t(t-1) \). This means multiplying \( t \) by each term inside the parentheses individually:

• Multiply \( t \) by \( t \), resulting in \( t^2 \).
• Multiply \( t \) by \( -1 \), resulting in \( -t \).

By applying the distributive property, \( t(t-1) \) becomes \( t^2 - t \). When we subtract this from \( t^2 \) as in the given function, it lays the groundwork for combining like terms later.

Like Terms

Like terms are pieces of an expression that contain the same variable raised to the same power. Identifying and combining like terms is crucial when simplifying expressions. When simplifying an expression like \( h(t) = t^2 - (t^2 - t) \), it's important to recognize and group the like terms. Here are the steps:

• Start by expanding or simplifying any expressions using properties such as the distributive property.
• In the expression, identify the terms \( t^2 \) and \(- t^2 \), which cancel each other out when you're subtracting \( t^2 - t^2 \).
• The remaining term, \( + t \), is left as it is since it has no like terms to combine with in this case.

Through these steps, \( h(t) \) is simplified to \( t \), making it easier to compare with other functions like \( g(t) \). Recognizing and combining like terms helps in determining the simplest form of a function.

Function Equivalence

Determining if two functions are equivalent involves simplifying and comparing their forms. In the given exercise, once both \( h(t) \) and \( g(t) \) are simplified, the process for checking function equivalence becomes straightforward.

• Simplify each function independently to its most reduced or simple form.
• For \( h(t) \), after applying the distributive property and combining like terms, we found it simplifies to \( t \).
• \( g(t) \) is already in its simplest form, which is also \( t \).
• Since \( h(t) \) and \( g(t) \) both simplify to \( t \), they are indeed equivalent.

Function equivalence is crucial, especially in algebra and calculus, as it allows you to confirm when two seemingly different expressions represent the same mathematical relationship. This concept is key in both simplifying complex functions and understanding the inherent relationships between different mathematical expressions.